POLAR COORDINATES
Up to this point, from Limits & Continuity through Derivatives and Integrals, for all
foundational calculus concepts, we've been working in cartesian coordinates.
You don't always hear them called cartesian coordinates, but we're talking about the points
in two-dimensional space that we graph on the xy-coordinate system.
We've got the x-axis and the y-axis, and the points we plot on these axes are defined using
the convention (x,y).
For example, if we have the point (4,3), it means that we move four units along the x-axis
in the positive direction, and three units along the y-axis in the positive direction.
The point we arrive at is (4,3).
So (x,y) is just a convention we use for expressing the location of a point in space.
But this isn't the only way we can express the location of this point.
Alternately, we could express its location as a distance away from the origin and an
angle between the point and the x-axis.
So, if I were giving you directions to the point (4,3), I could do it with cartesian
coordinates, telling you to walk in straight lines, left and right parallel to the x-axis,
and then up and down parallel to the the y-axis, or I could do it with polar coordinates, pointing
you in a certain direction, and then telling you how far to walk.
Where cartesian coordinates are expressed as (x,y), polar coordinates are expressed
as (r,theta), where r is the distance you walk from the origin, or the radius, and theta
is the direction angle in which I point you.
CONVERTING POLAR POINTS
If you think about it, you can easily convert cartesian coordinates to polar coordinates
with some simple trigonometry.
We already have a right triangle and we can call the lengths of the three sides x, y and
r, which means we can use the pythagorean theorem to express r in terms of x and y.
We'd just say that x^2+y^2=r^2.
Then, given any cartesian coordinate point, like our point (4,3), we can use this conversion
formula to solve for the r-value in the corresponding polar coordinate point.
Keep in mind that we're talking about distances here, so we can discount the negative value
of r.
Now that we have the r-value, we just need the corresponding value of theta, and then
we'll know how to express the cartesian point (4,3) as a polar point.
So let's talk about our angle theta.
Given our point in cartesian coordinates, we already know the length of the opposite
and adjacent sides of the triangle.
To generalize, we'll call them x and y.
Now remember SOH CAH TOA from trigonometry?
Since tangent of our angle is equal to the length of the opposite side divided by the
length of the adjacent side, we can take the inverse tangent of both sides to solve for
theta.
Now we can say that the cartesian point (4,3) is equal to the polar point (5,tan^{-1}(3/4)),
and we know how to use formulas to convert any cartesian coordinate to a polar coordinate.
CONVERTING POLAR EQUATIONS
Now let's talk about how to convert more than just points from cartesian to polar.
We can also use our right triangle to write expressions for x and y.
Since cosine of the angle is the length of the adjacent side divided by the length of
the hypotenuse, we can describe the relationship between r, theta and x, and solve this equation
for x.
We can do the same thing for y, knowing that sine of the angle theta is the length of the
opposite side divided by the length of the hypotenuse.
At this point you have everything you need not just to convert single points from cartesian
to polar, but to convert entire functions from cartesian to polar.
Given a function in terms of x and y, you can use these substitution formulas to remove
x and y, replacing them with r and theta, and before you know it you have an equation
defined in polar coordinates.
POLAR COORDINATES AND THE UNIT CIRCLE
Before we move on, we need to briefly discuss the unit circle and make a note about our
last example.
First, it's important for us to recognize the close relationship between the unit circle
and the polar coordinate system.
Remember that the unit circle gives us values for the angle theta as we move away from the
x-axis in a counterclockwise direction.
Conveniently, these angles match the angles that define the polar coordinate system, which
means that, given the polar point (2,pi/2), the angle pi/2 takes us up to 90 degrees,
and out a radius of 2 from the origin.
You'll be using the unit circle constantly to find values of theta in polar coordinates.
POSITIVE AND NEGATIVE VALUES IN POLAR COORDINATES
Now for the note about our last example.
We said that we could ignore the negative value of r, such that we could say that r
is just 5 instead of +/-5.
We did this because, even though r and theta can both assume negative values in polar coordinates,
if we want to we can deal entirely in positive values.
That's because there are always four ways to arrive at the same polar point.
For example, if we want to plot the polar point (5,pi/3), we already know that we find
our pi/3 angle and then move out a distance of 5 units from the origin.
To plot the same point we could instead find the angle 4pi/3 and move out in the opposite
direction a distance from the origin of -5.
Or we can move clockwise instead of counterclockwise to find theta, our angle will be -5pi/3 and
we'll move out from the origin a distance of 5.
Or, moving in the clockwise, negative theta direction again, we can take an angle of -2pi/3
and move out a distance of -5 from the origin.
Which means that the four polar points we've listed here are all the same point in space,
and given the point (-5,-2pi/3), we could decide we'd rather not deal with negative
signs and instead call this point (5,pi/3).
WHY USE POLAR?
Now you might be wondering: Why would you want to use polar coordinates when cartesian
coordinates were working so well for you?
Well, there are lots of functions that are much easier to express in polar coordinates
than in cartesian coordinates.
For example, think about the equation of a circle as you know it today.
The equation of a circle with radius 2, described in terms of x and y, looks like this (x^2+y^2=4).
But described in terms of polar coordinates, the equation of the exact same circle looks
like this (r=2).
And that's muuuuchhhh better!
Think about circles, spheres, and cylinders.
These are all much easier to describe in polar coordinates than they are in cartesian coordinates,
so it's very convenient to understand polar coordinates and be able to work with them
when you need to.